Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

The classic Cayley identity states thatdet(∂)(detX)s=s(s+1)⋯(s+n−1)(detX)s−1det(∂)(detX)s=s(s+1)⋯(s+n−1)(detX)s−1 where X=(xij)X=(xij) is an n×nn×n matrix of indeterminates and ∂=(∂/∂xij)∂=(∂/∂xij) is the corresponding matrix of partial derivatives. In this paper we present straightforward algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann–Berezin integration. Among the new identities proven here are a pair of “diagonal-parametrized” Cayley identities, a pair of “Laplacian-parametrized” Cayley identities, and the “product-parametrized” and “border-parametrized” rectangular Cayley identities.